Circular L(j,k)-labeling numbers of square of paths

Let j, k and σ be positive numbers, a circular σ-L(j, k)-labeling of a graph G is a function f : V (G) → [0, σ) such that |f(u) - f(v)|σ ≥ j if u and v are adjacent, and |f(u) - f(v)|σ ≥ k if u and v are at distance two, where |a - b|σ = min{|a - b|, σ - |a - b|}. The minimum σ such that there exist a circular σ-L(j, k)-labeling of G is called the circular-L(j, k)-labeling number of G and is denoted by σj,k(G). The k-th power Gk of an undirected graph G is a graph with the same set of vertices and an edge between two vertices when their distance in G is at most k. In this paper, the circular L(j, k)-labeling numbers of P2n are determined.